An embodiment of the present invention relates to a method for determining a radial refractive-index profile of a cylindrical optical object, particularly a preform for an optical fiber. The cylinder optical object comprises a cylinder longitudinal axis around which at least one layer k with a layer radius rk and with a layer refractive index nk extends radially symmetrically. A deflection angle distribution Ψ(y) is measured and the refractive index profile is reconstructed therefrom on the basis of a model.
Such cylindrical optical objects are, for instance, fiber preforms, optical fibers, light guides, or cylinder lenses. One of the important properties of such objects is their refractive index and the spatial distribution thereof, particularly the radial refractive-index distribution, which will also be called “refractive index profile” hereinafter. For instance, the refractive index profile of the fiber preform defines the waveguide characteristics of the optical fiber drawn therefrom. The optical objects of relevance in this case have a homogeneous or a stepped refractive-index profile. These are particularly optical preforms with a step index profile in the case of which a core with a higher refractive index is surrounded by at least one cladding layer with a lower refractive index.
The refractive index distribution, however, cannot be measured directly. Therefore, it is normally determined indirectly as a deflection or interference of a light beam which is transmitted through a volume region of the optical element, the stepwise transmission being also called “scanning” hereinafter. The real cause, i.e. the spatial refractive-index distribution in the optical element, can be inferred from the interference or the deflection of the exiting light beam (exit beam) based on the beam direction at the beam entrance point (entry beam). The family of the deflection angles measured during scanning of the light beam in a direction transverse to the cylinder longitudinal axis (in y-direction) is herein also called “deflection angle distribution” Ψ(y). For a better view and illustration, the geometric relationships are schematically shown in FIG. 3. The deflection angle Ψ is defined as the angle between exit beam 33 and entry beam 32, and y is defined as the distance between the cylinder longitudinal axis L and the entry point E of the entry beam 32. For radially symmetric objects with a step index distribution of the refractive index, it can be described mathematically with reference to the following formula (1):
                                          Ψ            m                    ⁡                      (            x            )                          =                  {                                                                                          2                    ·                                                                  ∑                                                  k                          =                          1                                                m                                            ⁢                                              [                                                                                                                                                                              arc                                  ⁢                                                                                                                                          ⁢                                                                      sin                                    ⁡                                                                          (                                                                                                                        y                                                                                      r                                            k                                                                                                                          ·                                                                                                                              n                                            0                                                                                                                                n                                                                                          k                                              -                                              1                                                                                                                                                                                                          )                                                                                                                                      -                                                                                                                                                                                                        arc                                ⁢                                                                                                                                  ⁢                                                                  sin                                  ⁡                                                                      (                                                                                                                  y                                                                                  r                                          k                                                                                                                    ·                                                                                                                        n                                          0                                                                                                                          n                                          k                                                                                                                                                      )                                                                                                                                                                                                      ]                                                                              ,                                                            for                                                                                                                                                                                      r                                                          m                              +                              1                                                                                ·                                                                                    n                                                              m                                +                                1                                                                                                                    n                              0                                                                                                      ,                                                                              r                                                          m                              +                              1                                                                                ·                                                                                                                                                                                                                              n                            m                                                                                n                            0                                                                          ≤                                                                            y                                                                          <                                                                              r                            m                                                    ·                                                                                    n                              m                                                                                      n                              0                                                                                                                                                                                                                                                                                                                                            2                          ·                                                                                    ∑                                                              k                                =                                1                                                                                            m                                -                                1                                                                                      ⁢                                                          [                                                                                                                                                                                                                  arc                                        ⁢                                                                                                                                                                  ⁢                                                                                  sin                                          ⁡                                                                                      (                                                                                                                                          y                                                                                                  r                                                  k                                                                                                                                            ·                                                                                                                                                n                                                  0                                                                                                                                                  n                                                                                                      k                                                    ⁢                                                                                                                                                                                                                  ⁢                                                    1                                                                                                                                                                                                                                        )                                                                                                                                                              -                                                                                                                                                                                                                                                  arcsin                                      ⁡                                                                              (                                                                                                                              y                                                                                          r                                              k                                                                                                                                ·                                                                                                                                    n                                              0                                                                                                                                      n                                              k                                                                                                                                                                      )                                                                                                                                                                                                        ]                                                                                                      +                                                                                                                                                                          2                          ·                                                      arccos                            ⁡                                                          (                                                                                                y                                                                      r                                    m                                                                                                  ·                                                                                                      n                                    0                                                                                                        n                                                                          m                                      ⁢                                                                                                                                                          ⁢                                      1                                                                                                                                                                  )                                                                                                      ,                                                                                                                        for                                                                                                                                                                                      r                                                          m                              +                              1                                                                                ·                                                                                    n                                                              m                                +                                1                                                                                                                    n                              0                                                                                                      ≤                                                                            y                                                                          <                                                                                                                                                                          r                                                      m                            +                            1                                                                          ·                                                                              n                            m                                                                                n                            0                                                                                                                                                                                                                    0                  ,                                                            for                                                                                                      y                                                        ≥                                      r                    1                                                                                                          (        1        )            where:m is the number of the layers of the objectn0 is the refractive index of the surrounding mediumnk is the refractive index of the k-th layerrk is the radius of the k-th layerA known mathematical method for calculating the refractive index profile from the “deflection angle distribution” Ψ(y) based on measurement data according to equation (1) is based on the so-called “Abel transform”.
                              n          ⁡                      [                          r              ⁡                              (                y                )                                      ]                          =                              n            0                    ·                      exp            ⁡                          (                                                1                  π                                ·                                                      ∫                    y                    R                                    ⁢                                                                                    Ψ                        ⁡                                                  (                          t                          )                                                                    ⁢                      dt                                                                                                                t                          2                                                -                                                  y                          2                                                                                                                                )                                                          (        2        )            where:r shortest distance from the cylinder longitudinal axis of the object to the beam path, namely:
                              r          ⁡                      (            y            )                          =                  y          ·                      exp            ⁡                          (                                                -                                      1                    π                                                  ·                                                      ∫                    y                    R                                    ⁢                                                                                    Ψ                        ⁡                                                  (                          t                          )                                                                    ⁢                      dt                                                                                                                t                          2                                                -                                                  y                          2                                                                                                                                )                                                          (        3        )            R reference point for the refractive index distribution, namely the radial position of the reference refractive index (atmosphere or index liquid outside the object), andΨ is substituted by δΦ/δt
U.S. Pat. No. 4,227,806 describes a method for non-destructively determining parameters of an optical fiber preform. The preform is scanned by means of a laser beam entering transversely into the core-cladding structure, and the deflection angle of the exiting beam is measured and subsequently compared to theoretical or empirical deflection angle distributions of preforms, whose refractive index distribution is known. During measurement, the preform is positioned in a bath containing immersion liquid so as to prevent the deflection angle from becoming too large.
U.S. Pat. No. 4,441,811 describes a method and an apparatus for determining the refractive index distribution of a cylindrical, transparent optical preform. In this case, too, the preform which is inserted in immersion liquid is scanned by a transversely entering light beam that extends perpendicular to the optical axis. The light beam is deflected by the glass of the preform and imaged with an optical device onto a positionable detector. The refractive index profile is calculated from the deflection angle distribution by way of numerical integration. Other preform parameters, such as preform diameter, core diameter, eccentricity and CCDR value (cladding to core diameter ratio) can also be determined therefrom.
Methods for the reconstruction of the refractive index profile from the transversely measured deflection angle distribution by using the Abel transform can also be found in U.S. Pat. Nos. 4,744,654, 5,078,488 and 4,515,475. The two following technical articles also describe such methods: MICHAEL R. HUTSEL AND THOMAS K. GAYLORD “Concurrent three-dimensional characterization of the refractive-index and residual-stress distributions in optical fibers”, APPLIED OPTICS, OPTICAL SOCIETY OF AMERICA, WASHINGTON, DC; US, Vol. 51, No. 22, 1 Aug. 2012 (Aug. 1, 2012), pages 5442-5452 (ISSN: 0003-6935, DOI: 10.1364/A0.51.005442) and FLEMING S. ET AL: “Nondestructive Measurement for Arbitrary RIP Distribution of Optical Fiber Preforms”, JOURNAL OF LIGHTWAVE TECHNOLOGY, IEEE SERVICE CENTER, NEW YORK, N.Y., US, Vol. 22, No. 2, 1 Feb. 2004 (Feb. 1, 2004), pages 478-486 (ISSN: 0733-8724, DOI: 10.1109/JLT.2004.824464).
The simple reconstruction of the refractive index profile n(r) from the transversely measured deflection angle distribution using the Abel transform according to above equation (2) does not, however, lead to negligible differences with respect to the real refractive-index profile. The reason for this is a known measurement artifact that occurs in refractive index discontinuities on boundaries between the transparent object and the environment or on the boundary between radial refractive-index steps. As shall be explained in more detail with reference to FIG. 2, measurements taken on the boundaries of refractive index jumps from a low to a high refractive index (when viewed from the outside to the inside) in a near-boundary volume region of the optical object lead to a region that can in principle not be measured. Typical differences and errors of the reconstructed refractive-index profile, for instance, of step index profiles are roundings of the profile and step heights that are too small. The technical article by Werner J. Glantschnig with the title: “Index profile reconstruction of fiber preforms from data containing a surface refraction component”; Applied Optics 29 (1990), July, No. 19, 2899-2907, deals with the problems posed by the non-measurable region. It is suggested that, by way of extrapolation based on the inner three measuring points of the deflection angle distribution directly before the discontinuity, the actually missing deflection angles are so to speak filled up in the non-measureable region.
The extrapolation based on three measuring points does not, however, produce good results in every case. To solve this problem, U.S. Pat. No. 8,013,985 B2 suggests a modification of this reconstruction method in that, for the measurement of the refractive index profile of a transparent cylindrical object such as a fiber preform, a beam deflection angle function is measured and the refractive index profile is reconstructed from the measured data on the basis of the paraxial ray theory mathematically and by application of an inverse Abel transform to the deflection function. In the measurement, the fiber preform to be measured is arranged between a laser and a transform lens. The preform has a central axis and a cylinder surface that define a preform radius R. The entry beam impinging on the cylinder surface at height x is deflected in the preform and exits again as an exit beam at another angle, which is detected by means of a photodetector and processed by a controller. The deflection angle is defined as the angle between the exit beam and the entry beam and is changed by varying the laser beam height x, and the deflection angle distribution ii is measured. An estimated refractive-index profile that is representative of the real refractive-index profile is adapted by means of a numerical model to the measured deflection angle distribution.
To this end, a symmetry correlation is carried out on the measured deflection function to define a center coordinate. The measured deflection function is split into two halves about the center coordinate, and a refractive index half-profile is calculated for each of the two halves to obtain a resulting estimated index profile for each half. The relevant parameters for the refractive index profile calculation are the preform radius R and the refractive index of the preform. A target angle distribution ψt is iteratively adapted to the measured deflection function, with measurement points close to a boundary (refractive index discontinuity) being omitted within or on the edge of the preform. This method of the arithmetical iterative adaptation of mathematical functions will also be called “fitting” in the following. According to U.S. Pat. No. 8,013,985, fitting is conducted in that the above equation (1) (however without consideration of the arccos portion indicated in the second line of the equation) has inserted thereinto yet unknown parameters of the refractive index profile, namely a value for the preform radius R (or for the radius of the refractive index discontinuity), as well as yet unknown refractive index values ηi, wherein the yet unknown parameters are varied such that the target angle distribution ψt obtained thereby best matches the measured deflection angle distribution ψm. The target angle distribution ψt is thus adapted (fitted) with the yet unknown parameters R and ηi to the measured deflection angle distribution ψm.
On the basis of the thus adapted, simulated target angle distribution ψt, a reconstructed refractive index profile η*i(r) is calculated. This profile extends up to the reconstructed preform radius R* which is greater than the radius RFIT of the inner object region. For cylindrical objects whose refractive index profile has at least one discontinuity, the method is applied to the various object regions which are respectively defined by the discontinuity.
In this method, a simulated target angle distribution ψt is adapted to the measured deflection angle distribution ψm by fitting yet unknown parameters, and a radial refractive index distribution which can extend up to the boundary of a further externally located discontinuity of the refractive index profile is calculated from the simulated target angle distribution.
The detection of a complete refractive-index profile of an optical object having several layers radially separated by a refractive index discontinuity therefore requires a successive measurement, calculation and estimation of the layers defined by the respective discontinuity from the outside to the inside. Systematic and numerical errors may result in both the fitting of the simulated target angle distribution ψt and in the conversion thereof into the reconstructed refractive-index profile η*i(r).
Moreover, it has been found that the comparison of deflection angle distributions, namely a simulated one and a measured one, is not very illustrative and requires a high degree of expertise for determining whether and optionally how a fitting is optimal, or whether and optionally which value requires a post-correction or further variation.
It is therefore an objective of the present invention to provide a method for determining the refractive index profile of a cylindrical transparent object with a radially symmetric or approximately radially symmetric refractive-index distribution, which is improved in terms of plausibility, accuracy and reproducibility.